Theorem rmoeqi 36552

Description: Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)

Hypothesis

Ref	Expression
rmoeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rmoeqi	⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)

Proof of Theorem rmoeqi

Step	Hyp	Ref	Expression
1		rmoeqi.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2856	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 633	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
4	3	mobii 2577	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5		df-rmo 3369	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-rmo 3369	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
7	4, 5, 6	3bitr4i 305	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃*wmo 2566  ∃*wrmo 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-mo 2568  df-cleq 2756  df-clel 2839  df-rmo 3369

This theorem is referenced by:  disjeq1i  36557